A372802 Number of partitions of [n] having exactly one block of maximal size and one block of minimal size (and any number of non-extremal blocks).

0, 1, 1, 4, 5, 16, 82, 169, 1381, 4162, 34346, 109099, 1114610, 5041271, 39441963, 269812729, 1972727781, 14983080612, 126099739072, 989666749503, 8839669627570, 79767000198673, 725399587976669, 6979798715649335, 69812296785011890, 703554021895986941

Offset

0,4

Comments

Minimal block and maximal block are identical if there is only one block.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..577

Wikipedia, Partition of a set

Example

a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
a(5) = 16: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(8) = 1381: 12345678, 1234567|8, 1234568|7, ..., 1|27|38|456, 18|2|37|456, 1|28|37|456.

Maple

b:= proc(n, i) option remember; `if`(n=i, 1, 0)+`if`(i<n, add(
      b(n-i*j, i+1)*combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i), 0)
    end:
a:= n-> signum(n)+add(binomial(n, i)*b(n-i, i+1), i=1..(n-1)/2):
seq(a(n), n=0..30);

Crossrefs

Cf. A000110, A007837, A224219, A372721.

Sequence in context: A127007 A007837 A032219 * A032144 A032049 A258063

Adjacent sequences: A372799 A372800 A372801 * A372803 A372804 A372805

Keyword

nonn

Author

Alois P. Heinz, May 13 2024

Status

approved